Point processes in random environment andapplication to the study of longevity risk
Sarah Kaakaı̈, Le Mans UniversitéCours Bachelier, 28/02/2020

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Some application of point processes§ Renewal of interest in point processes in the past years.
§ Flexibility allows for the modeling of a wide range of phenomena in:
Finance and Insurance (Dassios and Zho (2011), Giesecke and Kim(2011), Bacry et al. (2015), Jaisson and Rosembaum (2015), El
Euch et. al. (2016)...)
Neurosciences (Reynaut Bouret et al. (2013), Chevallier et al.
(2015), Galves and Löcherbach (2016)...)
Individual-based model in biology and ecology (Fournier and Méléard(2004), Champagnat et al. (2006), Méléard and Tran (2010), Billard
et al (2016)...)
Chemical reactions (Andersen and Kurtz (2015))
Epidemiology, cyber risk...

§ Human longevity?
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Point processes and longevity§ Observed data: population data (birth, death count).
ë longevity indicators (life expectancy, death rates,...) are aby-product of the population dynamics.
§ Individual-based model for human populations (Bensusan (2010),
Boumezoued and El Karoui (2016)).
§ Issues:
Non-stationarity, influence of macro-environment (El Karoui et al.
(2018)).
High heterogeneity: Structured population.
Interactions ñ non-linearity.

§ Need for probabilistic tools to deal with this complexity.

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Point processes§ General theory on point processes measures : 60-70’s.
§ Several viewpoint to define point processes:
Random counting measures (static).
Random sets (static).
In certain cases: Counting processes (multivariate, marked),
Dynamic viewpoint.

§ Central concept: Intensity (measure).
But not sufficient to defined point processes in general settings.
§ Pathwise construction.

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1 Some generalities on point processes
2 Pathwise representation of point processes in random environment

Process with bounded intensity
General caseA first “converse result”
3 Strong comparison of point processes

Point processes with ordered intensities
General case
Converse result
4 Birth-Death-Swap process in random environment

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Outline

1 Some generalities on point processes
2 Pathwise representation of point processes in random environment
3 Strong comparison of point processes
4 Birth-Death-Swap process in random environment

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Notations§ Filtered probability space pΩ, pGt q, Pq.
§ E Polish space and E its Borel σ-algebra.
§ PpGt q σ-algebra of predictable processes generated by processes
Ct “ H1st0 ,t1 s ptq,t ě 0, H P Gt0 .

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Random counting measure
Definition (Random counting measure/Point process (simple))
A random counting measure is a random measure M : Ω ˆ E Ñ N̄ suchthat Mω : A P E ÞÑ Mpω, Aq is a purely atomic and its every atom has

weight one, a.s.

§ Equivalent viewpoint: M is a r.v taking values in the space ofcounting measures.
§ Mean measure of M:µpAq “ ErMpAqs.
§ For any nonnegative measurable function f (f P E` ),żf pxqMpω, dxq.

Mpf qpωq “E

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Some properties of random measures§ Example M is a Poisson random measure with measure µ if:@A P E, MpAq „ PpµpAqq
if A1 , . . . , Ap are disjoints then MpA1 q, . . . , MpAp q are independent.

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Some properties of random measures§ Example M is a Poisson random measure with measure µ if:@A P E, MpAq „ PpµpAqq
if A1 , . . . , Ap are disjoints then MpA1 q, . . . , MpAp q are independent.

§ Proposition 1 The probability law of a random (counting)measure M on pE , Eq is uniquely determined by its Laplace
functional,f P E` .
şPoisson measures: Erexpp´Mpf qqs “ expp´ p1 ´ e f qdµq.
Erexpp´Mpf qqs,

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Some properties of random measures§ Example M is a Poisson random measure with measure µ if:@A P E, MpAq „ PpµpAqq
if A1 , . . . , Ap are disjoints then MpA1 q, . . . , MpAp q are independent.

§ Proposition 1 The probability law of a random (counting)measure M on pE , Eq is uniquely determined by its Laplace
functional,f P E` .
şPoisson measures: Erexpp´Mpf qqs “ expp´ p1 ´ e f qdµq.
Erexpp´Mpf qqs,

§ Proposition 2 The probability law of a random counting measure
M on pE , Eq is uniquely determined by the set of avoidance
probabilities:

PpMpAq “ 0q,A P E.

If PpMpAq “ 0q “ e ´µpAq with µ sigma-finite, then M is a Poisson
measure.
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Point process viewpoint§ The point process associated with a sequence pXi q of pE , Eq-randomvariables is defined for all A P E byÿ

MpAq “ÿ

1A pXi q,i1

T3 + (1T3 + (1U1

T1

T2 + (11

T3 + (1T2 + (1T1
T3 )V2T3 + (1U1

T3 )V2T3 + (1T5 + (1T5 + (1

1 T3 )V2

T2 )V1

U1

T3 )V2
T5 )V3

T5 + (1T5 )V3T5 )V3

U2

T4

U1
T1

T5 + (1T5 + (1T5 )V3

T5 + (1T5 )V3T1+ (1T2 + (1 T
U32 TT24)V1

T2 )V1

T3 )V2T3 + (1T1

T3 + (11U1

1T3 )V2

T3 )V2

U1

T1

1

U1

i

1

1

1
U1

δXi q.

pM “

T3 + (1T3 )V2
T5 + (1T5 )V3T3 )V2T2 + (1

U21 T4
T2 )V

ϕt

(1
U2T5T+
4T5 )V3

T1
T5 )V3

T5 + (1T5 )V3
T2 )V1U2 T4ccc
T2 + (1T2 )V1T= 1Tc2X
T2 = T1X
T = T2X (1T5 = T3X)Vc
T4
X3 T2 +
X
2
T
=
T
T
=
T
T
= T2X U T4 = Tc2X
1
2
3
1c
1
c
2 cT4
X
X
X5 = T X
XcT )V2
T4T3=+TT2(1T2 = T1X
T3 =1 T2X
T2 = TT
3T3 = T2
1 = T1 3
1
T2 + (1

T1

U1

T1

T1 = T11X

T2 + (1T1 = T1X
T2 )V1T2 + (1T2 )V11
T1 = T1X

U2

T3 + (1T4c

T3 = T2X
U 1 T1U1 T cT2 = T11X
T1 = T1Xc
T4 = T2X
T3 = T2Xcc
T2 = T1X
T3 = T2X

T4 = T2X

T2 + (1T2 )V1

T4 = T2X

2

U2

T4

T5 = T3Xc

T5 + (1T4 = T2Xc
T5 = T3X
T3 )V2cT5 )V3

T5 = T3Xc

T5 + (1

T4

T1

T4

§ Random set viewpoint: Mpωq “ tx P E ; x “ Xi pωq for some iu.
T1 = T1X

T5 = T3X

T5 )V3cc

1
T1 c= T1X
T2 = T1X X T3 = T2X
X c + (1)V
T4 =
T2 = T1X
T3 = T2T
T5T
23
5
T2 )V1T3 + (1U2 T4Xc
T5 = T3
T2 + (1 X c T2 )V1
T5 = T3
U
U1

T2 + (1T5 = T3X
c

U2

T4 = T2X
T5 + (1U

2 T4
U 1 T1Xc
T3 + (1 T2 =
T3T)V
1 2

T1 = T1X
T1 = T1X

T3 )V2T2 = T1Xc

T3 = T2Xc

T2 + (1T4 = T2X

T2 )V1

T5 = T3X

c

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T5 )V

Point process viewpoint
The point process associated with a sequence pXi q of pE , Eq-randomvariables is defined for all A P E by
MpAq “ÿ

1A pXi q,

ÿ
pM “

i

δXi q.

işř§ Mpf q “ E f pxqMpω, dxq “ i f pXi q.

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Point process viewpoint
The point process associated with a sequence pXi q of pE , Eq-randomvariables is defined for all A P E by
MpAq “ÿÿ

1A pXi q,

pM “iδXi q.

işř§ Mpf q “ E f pxqMpω, dxq “ i f pXi q.
§ Proposition: If M is a random counting measure and µ is σ-finite,M“ÿδ Xi ,iwith pXi q a sequence of r.v taking values on a enlargement pĒ , Ēq ofpE , Eq.

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Point processes on the half line
Case E “ R` .
§ Atoms can be ordered: pXi q Ñ increasing sequence pTn qnPN .
§ Associated counting process
Nt “ Nps0, tsq “8ÿ

1tTn ďtu .

n“1(Source: Bremaud)

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Point processes on the half line
Case E “ R` : dynamic viewpoint.
§ Associated counting process
Nt “ Nps0, tsq “8ÿ

1tTn ďtu .

n“1§ The random measure is seen as a r.v taking values in the space A ofcounting functions.

§ If N is adapted to a filtration pGt q, for C P PpGt q
żt
Cs dNs “0ÿ

CTn 1tTn ďtu

ně0

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Poisson process§ Dynamic viewpoint Let N be a counting process adapted to afiltration pGt q. Then N is a pGt q-Poisson process if:(i) Independent increments:(ii) Stationary increments:Ns`h ´ Ns K
K Gs .
dNpss, s ` hsq “ Ns`h ´ Ns “ Nh p„ Ppλhqq.
§ Point process viewpoint
8ÿN“

1tTn ďtu ,n“1pSn qně1 “ pTn ´ Tn´1 qně1 i.i.d „ Epλq.

§ Random measure viewpoint For any f P E` ,żEre

´Npf qp1 ´ e ´f psq qνpdsqq,

s “ expp´ν “ λ Leb.

R`
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Intensity and martingale property
Let N be a pGt q Poisson process with intensity λ.
§ Let Cs “ 1H 1st0 ,t1 s psq with H P Gt0 . Then,ż8
ż8Er
Cs dNs s “ Er1H pNt1 ´ Nt0 qs “ Er1H sλpt1 ´ t0 q “ ErCs λdss
0

0

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Intensity and martingale property
Let N be a pGt q Poisson process with intensity λ.
§ Let Cs “ 1H 1st0 ,t1 s psq with H P Gt0 . Then,ż8
ż8Er
Cs dNs s “ Er1H pNt1 ´ Nt0 qs “ Er1H sλpt1 ´ t0 q “ ErCs λdss00§ Generalization to nonnegative predictable processes C P PpGt q:
ż8ż8Er
Cs dNs s “ ErCs λdss.
0(1)0§ Equivalent property:
pNt ´ λtq is a pGt q-martingale.
§ (1) can be generalized to define the intensity process of a counting
process.
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General case: stochastic intensity
Definition (Stochastic intensity (Bremaud))
Let N be an adapted counting process, and pλt q a nonnegativeżt
pGt q-process with@t ě 0,λs ds ă 8a.s..

(2)0Then, N admits the pGt q-intensity pλt q if for all nonnegative predictableprocess C
ż8
ż8
Cs dNs s “ ErCs λs dss.
Er
0

(3)

0

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General case: stochastic intensity
Definition (Stochastic intensity (Bremaud))
Let N be an adapted counting process, and pλt q a nonnegativeżt
pGt q-process with@t ě 0,λs ds ă 8a.s..

(2)0Then, N admits the pGt q-intensity pλt q if for all nonnegative predictableprocess C
ż8
ż8
Cs dNs s “ ErCs λs dss.
Er
0

(3)

0

§ (2) is a non-explosion condition.
ş§ Equivalenty: pNt ´ 0t λs dsq is a pGt q local martingale.
§ The existence of an intensity has to do with the absolute continuityof the predictable compensator A of N w.r.t Leb. If λ exists,ştAt “ 0 λs ds.
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Examples and uniqueness§ Non-Homogeneous Poisson processes (NHP) pλt q “ pf ptqq
is a deterministic function.
§ Cox/Doubly stochastic Poisson processes
pλt q is G0 -measurable.
(Nt ´ Ns is independent of Gs given G0 )§ Uniqueness of intensity process In order to be unique, pλt q should
be taken predictable.

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Examples and uniqueness§ Non-Homogeneous Poisson processes (NHP) pλt q “ pf ptqq
is a deterministic function.
§ Cox/Doubly stochastic Poisson processes
pλt q is G0 -measurable.
(Nt ´ Ns is independent of Gs given G0 )§ Uniqueness of intensity process In order to be unique, pλt q should
be taken predictable.
§ Does the intensity process characterizes the probability distributionof a counting process?

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Martingale characterization
Let us come back to the example of the Poisson process:
Theorem (Martingale characterization of Poisson processes (Watanabe))
Let N be a counting process and f a locally integrable nonnegativeşt
function such that N ´ 0 f psqds is pGt q-martingale.
Then N is a pGt q-non homogeneous Poisson process of intensity functionf.
§ Extension The result still holds for Cox processes (Bremaud)§ Sketch of the proof:ştiu

Use (3) to show that pZt q “ pe iuNt ´ 0 pe ´1qf psqdsq q is apGt q-martingale.
Laplace functional characterization.

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General case(I)Does the intensity process characterizes the probability distribution of acounting process?

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General case(I)Does the intensity process characterizes the probability distribution of acounting process?
ë NO without further assumptions.
§ Intensity process is often a function of the counting process itself.

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General case(I)Does the intensity process characterizes the probability distribution of acounting process?
ë NO without further assumptions.
§ Intensity process is often a function of the counting process itself.
§ Examples
Pure Birth process with immigration: λs “ a ` bNs .
Such a process is a Continuous Time Markov Chain (CTMC).
T1 , T2 ´ T1 , T3 ´ T2 , . . . are independent and Tk`1 ´ Tk „ Epa ` kbq.
(Linear) Hawkes process:żs
λs “ a `hps ´ r qdNr .
0şsλs “ f prNss ´ q, with rNss “ pNt^s qtě0 and f : n P A ÞÑ pa ` 0 hps ´ r qdnpr qq.
In particular, if pf prNst ´ qq is the pGt q-intensity of N then pf prNst ´ qq
is also its pFtN q intensity.
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General case(II)Theorem (Jacod 75 (partial))
Let λ : Ω ˆ R` ˆ A Ñ R` a pGt q-predictable functional such that for all

counting path n P A and t ě 0, λpω, t, rnsq P G0 .

Then if two counting processes N and N 1 have respective pGt q intensitiespλpω, t, rNst ´ qq and pλpω, t, rN 1 st ´ qq, then N and N 1 have the same
distribution.
§ Also weak existence result (under additional assumptions).

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General case(II)Theorem (Jacod 75 (partial))
Let λ : Ω ˆ R` ˆ A Ñ R` a pGt q-predictable functional such that for all

counting path n P A and t ě 0, λpω, t, rnsq P G0 .

Then if two counting processes N and N 1 have respective pGt q intensitiespλpω, t, rNst ´ qq and pλpω, t, rN 1 st ´ qq, then N and N 1 have the same
distribution.
§ Also weak existence result (under additional assumptions).
BUT
Random environment/external noise is G0 -measurable “known at time 0”.
ë Stochastic intensity not sufficient to introduce a counting process.

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Space-time point process
Result originally stated for Marked point processes.

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Space-time point process
Result originally stated for Marked point processes.
§ Space-time point process
Random counting measure N on R` ˆ E .
Defined relatively to pG
1 t q if for all A P E, Npr0, ts ˆ Aq P Gt
1

T3 + (1✓<latexit sha1_base64="JqEnYvV6PtsKBJYmBVwEpjIMANw=">AAAB7XicbVDLSgNBEJyNrxhfUY9eBoPgKeyKoMegF48RzAOSJcxOOsmY2ZllplcIS/7BiwdFvPo/3vwbJ8keNLGgoajqprsrSqSw6PvfXmFtfWNzq7hd2tnd2z8oHx41rU4NhwbXUpt2xCxIoaCBAiW0EwMsjiS0ovHtzG89gbFCqwecJBDGbKjEQHCGTmp2cQTIeuWKX/XnoKskyEmF5Kj3yl/dvuZpDAq5ZNZ2Aj/BMGMGBZcwLXVTCwnjYzaEjqOKxWDDbH7tlJ45pU8H2rhSSOfq74mMxdZO4sh1xgxHdtmbif95nRQH12EmVJIiKL5YNEglRU1nr9O+MMBRThxh3Ah3K+UjZhhHF1DJhRAsv7xKmhfVwK8G95eV2k0eR5GckFNyTgJyRWrkjtRJg3DySJ7JK3nztPfivXsfi9aCl88ckz/wPn8Ao/ePKA==</latexit><latexit

T3 + (11

T3 )V2T3 + (1T3 )V2

U1

U1
T1

T5 + (1T3 )V2T3 + (1

1

T1

1
U1

T2 + (1T2 )V1T2 + (1T2 )V11= T1X

T2 + (1T1U2c
T4 = T2X
T3 = T2Xcc
T2 = T1X
T3 = T2X

T1 = T1X
T2 + (1T2 )V1

T5 + (1T2 )V1
T3 )V2

U2
U1

T2 + (1T4 = T2X

T2 = T1X

T1 = T1Xc

T5 + (1T2 )V1

U2

T5 )V3

U2

T4
T3 )V2

T5 + (1ϕtcT4 = T2Xc
T5 = T3X
T3 )V22

U2

T5 )V3
T5 + (1c1T1 c= T1X
T2 = T1X X T3 = T2X
X c + (1)V
T4 =
T2 = T1X
T3 = T2T
T5T
23
5
T2 )V1T3 + (1U2 T4Xc
T5 = T3
T2 + (1 X c T2 )V1
T5 = T3
U
c

T4

T4

T1

T3 = T2X

T4 =UT12XT1

T5 = T3X

c

T4

t

T4

T2 = T1X

T5 )V3T5 )V3T2 + (1T3 + (1T1 = T1X
U1 T cT2 = T11X

T5 + (1T5 )V3T4

U2 T4
U1 TT21+ (1T2 )V1T3 + (1T3 )V2

T3 )V2

T1

T3 + (11

T3 + (1T3 )V2<latexit sha1_base64="fInOqGTCCrWkRGJFOZWK1l6FLBY=">AAAB6HicbVBNS8NAEJ34WetX1aOXYBE8lUQEPRa9eGzBfkAbymY7adduNmF3IpTSX+DFgyJe/Une/Ddu2xy09cHA470ZZuaFqRSGPO/bWVvf2NzaLuwUd/f2Dw5LR8dNk2SaY4MnMtHtkBmUQmGDBElspxpZHEpshaO7md96Qm1Eoh5onGIQs4ESkeCMrFSnXqnsVbw53FXi56QMOWq90le3n/AsRkVcMmM6vpdSMGGaBJc4LXYzgynjIzbAjqWKxWiCyfzQqXtulb4bJdqWIneu/p6YsNiYcRzazpjR0Cx7M/E/r5NRdBNMhEozQsUXi6JMupS4s6/dvtDISY4tYVwLe6vLh0wzTjabog3BX355lTQvK75X8etX5eptHkcBTuEMLsCHa6jCPdSgARwQnuEV3pxH58V5dz4WrWtOPnMCf+B8/gDgKYz4</latexit><latexitcT3 = T2Xc

T2 + (1T4 = T2X

T2 )V1

T5 = T3X

c

U

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Marked point process§ Space-time point process
Random counting measure N on R` ˆ E .
Defined relatively to pGt q if for all A P E, Npr0, ts ˆ Aq P Gt
§ Marked point process on pR` ˆ E , BR` ˆ Eq.
Nt pAq “ Npr0, ts ˆ Aq “ÿ

1tTn ďtu 1A pXn q,nwith pTn q increasing sequence and pXn qně0 sequence of r.v
taking values in the mark space E .
In particular, pNt pE qq is a counting process.
§ All space-time point processes are not Marked point processes!

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Examples (Marked Point processes)§ Multivariate counting process E “ tx1 , . . . , xp u
Nti “ Nt ptiuq “ÿ

1tTn ďtu 1tXn “xi u ,i “ 1, ..., p.

nPure birth marked by discrete frailty variables, interacting Hawkes
processes...
Usual hypothesis: components of pN 1 , ..., N p q have no jumps incommon.

§ Compound Poisson process pTn q are jump times of a Poissonprocess, marks pXi q are i.i.d „ π and K
K of pTn q.
A compound Poisson process is a Poisson random measure of mean
measure λds b π.

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pGt q Poisson measures (I)Let γ be a sigma-finite measure on pE , Eq.
Definition (Space-time pGt q Poisson measure)
Qpdt, dxq is a pGt q-Poisson measure on R` ˆ E of mean measure
dt b γpdxq iff @A1 , ..., An disjoint sets with γpAi q ă 8, the counting
processes Qt pAi q defined for i “ 1..p by
Qt pAi q “ Qpr0, ts ˆ Ai q,@t ě 0are independent Gt -Poisson processes of intensity γpAi q.
§ If E “ txu, Q is a pGt q-Poisson process of intensity γptxuq.
§ When γ is finite, Q “ tpTn , Xn qu is a compound Poisson process. It
particular, its jump times can be enumerated increasingly.
It is not the case when γ is only sigma-finite.
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pGt q Poisson measures (II)Definition (Space-time pGt q Poisson measure)
Qpdt, dxq is a pGt q-Poisson measure on R` ˆ E of mean measure
Lebpdtqγpdxq iff @A1 , ..., An disjoint sets with γpAi q ă 8, the counting
processes Qt pAi q defined for i “ 1..p by
Qt pAi q “ Qpr0, ts ˆ Ai q,@t ě 0are independent Gt -Poisson processes of intensity γpAi q.
ErQt pAqs “ tγpAqż
1r0,tsˆA ps, xqQpds, dxqs “ ErżEr

ô
1r0,tsˆA ps, xqγpdxqdss.

R` ˆE

R` ˆE

§ Generalization to predictable processes pG ps, xqq P PpGt q b E
ż
Er
R` ˆE

ż
G ps, xqQpds, dxqs “ ErG ps, xqγpdxqdss.

R` ˆE
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Stochastic intensity
Stochastic intensity of space-time point process N
Predictable random measure λpω, s, dxq such that for all G P PpGt q b EżErżG ps, zqNpds, dzqs “ ErR` ˆE

G ps, zqλps, dxqdss.

(4)

R` ˆE

Equivalently,ş
The counting process pNt pAqq has the pGt q intensity pλt pAqq “ p A λpt, dxqq.
§ pGt q intensity of Poisson random measure : λps, dxq “ γpdxq.
§ The pGt q multivariate intensity of a multivariate counting process
N “ pN 1 , ..., N p q is the vector λ “ pλ1 , ..., λp q of the components
pGt q-intensity processes.
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Characterization of space-time point processes§ A Poisson measure is uniquely characterized by its intensity.
§ If Gt “ G0 _ σpNs pBq : s ď t, B P Eq, then two marked point

processes with pGt q intensity pλpt, dxqq have the same distribution.

But§ Poor structure random environment/external noise which is
G0 -measurable.
§ Existence results in distribution, no pathwise construction.

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Outline

1 Some generalities on point processes
2 Pathwise representation of point processes in random environment
Process with bounded intensity
General caseA first “converse result”
3 Strong comparison of point processes
4 Birth-Death-Swap process in random environment

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Pathwise representation of point processes
Two main approaches:§ Multiple random time changes (Kurtz (1980), Garcia (1995), Garcia
and Kurtz (2008)).
§ Thinning and projection of Poisson measures
Population dynamics (Fournier and Méléard (2004), Garcia andKurtz (2006), Méléard and Tran (2009), El Karoui and Boumezoued(2016)), interacting Hawkes processes (Chevallier et. al (2015),
Delattre et al (2016),..), PDP (Lemaire et al. (2018)...)
General construction in the spirit of Massoulié (1998): Point
processes described by an intensity process+ Poisson measure.

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Pathwise representation of point processes
Two main approaches:§ Multiple random time changes (Kurtz (1980), Garcia (1995), Garcia
and Kurtz (2008)).

§ Thinning and projection of Poisson measures
General construction in the spirit of Massoulié (1998): Point
processes described by an intensity process+ Poisson measure.

First step: How to simulate a counting process with bounded intensity
λt ď λ̄?

ş λ§ Naive idea: take Ntλ “ 0t s dNs with N Poisson process of intensityλ̄ştλ̄. pNtλ ´ 0 λs dsqtě0 is a local martingale.

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Pathwise representation of point processes
Two main approaches:§ Multiple random time changes (Kurtz (1980), Garcia (1995), Garcia
and Kurtz (2008)).

§ Thinning and projection of Poisson measures
General construction in the spirit of Massoulié (1998): Point
processes described by an intensity process+ Poisson measure.

First step: How to simulate a counting process with bounded intensity
λt ď λ̄?

ş λ§ Naive idea: take Ntλ “ 0t s dNs with N Poisson process of intensityλ̄ştλ̄. pNtλ ´ 0 λs dsqtě0 is a local martingale.
But N λ is not a counting process.

§ Solution: Increase space dimension and represent space-time point
processes as strong solutions of SDEs driven by Poisson measures.
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Outline
1 Some generalities on point processes
2 Pathwise representation of point processes in random environment
Process with bounded intensity
General caseA first “converse result”
3 Strong comparison of point processes
Point processes with ordered intensities
General case
Converse result
4 Birth-Death-Swap process in random environment

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Thinning with bounded intensity
Simulation of counting process N with given (stochastic) bounded
intensity λt ď λ̄ (Lewis and Shedler (1979)):

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Thinning with bounded intensity
Simulation of counting process N with given (stochastic) bounded
intensity λt ď λ̄ (Lewis and Shedler (1979)):
1

Take Q Marked Poisson measure on R` ˆ r0, 1s with intensity
λ̄ds b du:Qpds, duq “řδTn psqδUn puq

ně1

with
pTn qně1 jump times of Poisson process of intensity λ̄.
pUn qně1 i.i.d variables on r0, 1s.
2

Accept Tn as jump time of N if
Un ď

λTn
.
λ̄

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NHP with intensity λt “ e ´βtStep 1: Simulation of tpTn , Un qně0 u.

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NHP with intensity λt “ e ´βtStep 2: Un ď

λ Tn
.
λ̄

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NHP with intensity λt “ e ´βtStep 2: Un ď

λ Tn
.
λ̄

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NHP with intensity λt “ e ´βtStep 3: Projection on R`

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Thinning equation: bounded case (I)
Nt “

ÿ
ně1

1tTn ďtu 1

tUn ď

ÿ
λT

n u
λ̄

“

1tTn ďtu 1D λ pTn , Un q,

ně1with D λ the predictable set:
D λ “ tps, uq; u ď

λs
u.
λ̄

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Thinning equation: bounded case (I)Nt “ÿ

1tTn ďtu 1

tUn ď

ně1

ÿ
λT

n u
λ̄

“

1tTn ďtu 1D λ pTn , Un q,

ně1with D λ the predictable set:
D λ “ tps, uq; u ď

λs
u.
λ̄

Thinning equation associated with simulation
Step 2 Projection of Q on D λ :Q D pds, duq “ 1D λ ps, uqQpds, duq “ř

1D λ pTn , Un qδTn psqδUn puq.

ně1

Step 3 Projection on R` :
żtż
Ntλ “

żtż
Q D pds, duq “

0

r0,1s

0

r0,1s

1tuď λs u Qpds, duq.
λ̄

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Thinning equation: bounded case (II)§ Stochastic intensity: Let C P PpGt qżtżtżλEr Cs dNs s “ Er
0

0

r0,1s

Cs 1tuď λs u Qpds, duqs
λ̄

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Thinning equation: bounded case (II)§ Stochastic intensity: Let C P PpGt qżtżtżλEr Cs dNs s “ Er
0

0

żtż
“ Er
0

r0,1s

r0,1s

Cs 1tuď λs u Qpds, duqsλ̄

Cs 1tuď λs u λ̄dudss
λ̄

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Thinning equation: bounded case (II)§ Stochastic intensity: Let C P PpGt qżtżtż
Er Cs dNsλ s “ Er
0

0

żtż
“ Er
0

r0,1s

żt
“ Er ds0

r0,1s

Cs 1tuď λs u Qpds, duqsλ̄

Cs 1tuď λs u λ̄dudss

˜

λ̄

ż λs
λ̄

Cs λ̄
0

¸

żt
du s “ Er Cs λs dss
0

34/66

Outline
1 Some generalities on point processes
2 Pathwise representation of point processes in random environment
Process with bounded intensity
General caseA first “converse result”
3 Strong comparison of point processes
Point processes with ordered intensities
General case
Converse result
4 Birth-Death-Swap process in random environment

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Thinninq equation: General case✓tSpace-time pGt q Poisson measure Q on R` ˆ R` of mean measure
dt b dθ.
No increasing enumeration of jumps times.
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Thinninq equation: General case
Given a predictable process pλt qtě0 withşt0

λs ds ă `8 a.s. @t ě 0(nonexplosion condition)✓✓={(t, ✓); ✓ 

t

t}

t
Restriction to predictable subset:
tps, θq; θ ď λs pωq, s ď tu
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Thinninq equation: General case✓✓={(t, ✓); ✓ 

T1

T2

t

t}

T3

T4 · · ·t“Theoretical” Thinningşt ş
Ntλ “ 0 R` 1tθďλs u Qpds, dθq is a counting process of pGt q-intensity λt .
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Some remarks§ General case: Thinning does not give a simulation procedure.
§ Bounded case: Q can replaced by 1tuďλ̄u Q Poisson measure ofintensity ds b 1tuďλ̄u du.
„ Compound Poisson process of mean measure λ̄ds b§ pλs q does not characterizes the distribution of N.

11du.
λ̄ tuďλ̄u

żtż
Ntλ “
0

R`

1tuďλs u Qpds, duq.

(5)

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Some remarks§ General case: Thinning does not give a simulation procedure.
§ Bounded case: Q can replaced by 1tuďλ̄u Q Poisson measure ofintensity ds b 1tuďλ̄u du.
„ Compound Poisson process of mean measure λ̄ds b§ pλs q does not characterizes the distribution of N.

11du.
λ̄ tuďλ̄u

żtż
Ntλ “
0

R`

1tuďλs u Qpds, duq.

(5)

§ Space-time point processes of pGt q intensity λps, dxq “ µps, xqγpdxq:żtżλ1tuďµps,xqu Qpds, du, dxq.

N pdt, dxq “
0

R` ˆE

with Q pGt q space-time Poisson measure on R` ˆ R` ˆ E of mean
measure ds b dθ b γpdxq.
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SDE driven by Poisson measure§ When λt “ αpω, t, rNst ´ q is a functional of N, (5) ñ SDE driven by
Q:
żtż
Ntα “
0

R`

1tθďαps,rN α ss ´ qu Qpds, dθq,

dNtα “ Qpdt, s0, αpt, rN α st ´ qsq§ Existence of a well-defined (non-exploding) solution?

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SDE driven by Poisson measure§ When λt “ αpω, t, rNst ´ q is a functional of N, (5) ñ SDE driven by
Q:
żtż
Ntα “
0

R`

1tθďαps,rN α ss ´ qu Qpds, dθq,

dNtα “ Qpdt, s0, αpt, rN α st ´ qsq§ Existence of a well-defined (non-exploding) solution?
If α is uniformly bounded by a constant λ̄: same as in bounded case(recursive procedure).
Even in simpler case α is not uniformly bounded.
Example: Linear birth process, λt “ bNt ´ .

§ In the following: Existence, uniqueness and nonexplosion by strong
domination.

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Outline
1 Some generalities on point processes
2 Pathwise representation of point processes in random environment
Process with bounded intensity
General caseA first “converse result”
3 Strong comparison of point processes
Point processes with ordered intensities
General case
Converse result
4 Birth-Death-Swap process in random environment

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“Converse” result (I)If a counting process N λ admits a pGt q-intensity pλt q,
is there a representation of N λ in terms of stochastic integral withrespect to some Poisson process?

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“Converse” result (I)If a counting process N λ admits a pGt q-intensity pλt q,
is there a representation of N λ in terms of stochastic integral withrespect to some Poisson process?
§ YES: see e.g. Grigelionis (1971), Jacod (1980), Massoulié (1998).
§ Ingredients:
A space-time Poisson measure Q̂pds, dθq of intensity ds b dθ andindependent of G8 .
A sequence of i.i.d variables pUn qně0 uniform on r0, 1s andindependent of G8 .

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“Converse” result (II)1
T3 + (1T3 )V2
T5 + (1U1

T2 + (1

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T5 )V3T1

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c

U2

c

T4

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T5

T4

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T6 T7

T5 = T3X

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41/66

“Converse” result (II)1
T3 + (1T3 )V2✓=

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U1

T2 + (1

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X
1<latexit sha1_base64="rbdbXszceVlCarz7ZPU3J4CJ62o=">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</latexit>ϕtT2 )V1T2 = T1X

c

U2

c

T4

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T5

T4

T4 = T2X<latexit sha1_base64="yI/kTzsucDEwDJpFXLuQhMB4Swo=">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</latexit>

T6 T7

T5 = T3X

c<latexit sha1_base64="Pja/6IRGgik86FHPrvK0hjPeN9w=">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</latexit><latexit sha1_base64="DXfzMfQeC50gpronrLctQJiz5vw=">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</latexit>

41/66

“Converse” result (II)1+ (1T3 )V2
3(1T3 )V2 1
1
1

T3 + (1

T1

U2

T

U1

T2 U24<latexit sha1_base64="Ufmet9rTZ93PA3FPF3rToZLhsjE=">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</latexit>

T2 + (1T2 )V1

UU

T23(1T43
T2 +<latexit sha1_base64="iOQxb+6xEieInwWXtmaovqUX9A4=">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</latexit>c

T4 = T2X
T3 =cT2X
X
T4 = T2X
3 = T2

T1X

cT3 )V2U 1 T1)V2
T3 + (1 T5 +
T3(1

T1

U1

U2 T4
U1 1 T1
TU
1<latexit sha1_base64="tM7q1As5LqO9ZzCSDrbLzpT2BZ0=">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</latexit>

T3 )V2T3 + (1T5 + (1T3 )V2

T3 )V2✓=

T5 + (1<latexit sha1_base64="9KzPIYMD8J7gdy1GKbBgQ62WgNA=">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</latexit>

T3 + (1

U1

)V1
T1

T5 + (1T )V3+ (1T )V3T53 )V2(1
3
T5 + T5
U 1 T1T3 + (11

T3 )V2

1

U1

T3 + (11T2 )V1T2 + (1T3 + (1T5 )V3T2 + (1T2 + (1T2 + (1T2 )V1U2

T2 + (1<latexit sha1_base64="LsxKoZ3ytmV5ml1dk6yxYiMjACY=">AAAC1XicjVHNSsNAGJzGv1r/oh69BIvgqSRS0GPRi8cKTVtoS0jSbQ1Nk5BsCqX0Jl59Aa/6SuIb6Fv47boFtYhuSDI738zsfrteEgYZN83Xgrayura+UdwsbW3v7O7p+wfNLM5Tn9l+HMZp23MzFgYRs3nAQ9ZOUuaOvZC1vNGVqLcmLM2COGrwacJ6Y3cYBYPAdzlRjq53QxL3XWfWcKpz26k6etmsmHIYy8BSoAw16rH+gi76iOEjxxgMETjhEC4yejqwYCIhrocZcSmhQNYZ5iiRNycVI4VL7Ii+Q5p1FBvRXGRm0u3TKiG9KTkNnJAnJl1KWKxmyHoukwX7W/ZMZoq9TenvqawxsRy3xP7lWyj/6xO9cAxwIXsIqKdEMqI7X6Xk8lTEzo0vXXFKSIgTuE/1lLAvnYtzNqQnk72Ls3Vl/U0qBSvmvtLmeBe7pAu2fl7nMmieVSyzYt1Uy7VLddVFHOEYp3Sf56jhGnXYlD3BI57wrLW0uXan3X9KtYLyHOLb0B4+AEIKla8=</latexit><latexit sha1_base64="qr2hWjwEUOI4JkONUXVF86o4RJM=">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</latexit>

T5 )V3
UT25 T+4 (1T2 )V1

T4

T2 )V1U2 T4
T 4 U 4 U2 T
c
c4
U
X
5
5
T1 = TT1X
T2 = TT1X
5 =cT3
T5 = T3X
T2 )V1

T5 + (1T2 )V1

T1
T2 )V1

T3 )V2<latexit sha1_base64="kCfPw5x1iGmL4NQJnGeAmg6hSoI=">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</latexit>t

T5 + (1T5 )V3<latexit sha1_base64="8TBF12fPSOyMG2XmMan/CjOaXxg=">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</latexit><latexit sha1_base64="zeO+XWYiVUNpTYsT6d9Biy4r/xY=">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</latexit><latexit sha1_base64="rbdbXszceVlCarz7ZPU3J4CJ62o=">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</latexit>

T4<latexit sha1_base64="yi5c0rp6wL2fELOTzUi9HZWJMmM=">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</latexit>

T5<latexit sha1_base64="yI/kTzsucDEwDJpFXLuQhMB4Swo=">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</latexit>

T5 )V3

T5 + (1

T5 )V

ϕt

U2

T4

U2 T4
T6 U 6
U2

T3 = T2Xc

T4

T4 = T2Xcc
T1 = T1X
T4 = T2X
T2 = T X
T3 = T2Xccc
T1 = T1X
T4 = T2X c 1
T2 = T1X
T3 = T2X c
T5 = T3X
X
X
X
X
T1 = T1
T4 = T2
T2 = T1 X c T3 = T2
X
Xc
T4 = T2
T3 = T2
T5 = T3
c
c
c
c
X Xc
T1 = T1X
T4T2=
T2X
T4T=
T2X
T2 = T1X T1 =T3T=
T1X
T3 = T2X
T3X
5 =
1c T2
c=
c
T1 = T1X
T4 = T2X
T2 = T1X
T3 = T2X
T5 = T3X

T1 T2 T3

T

T5 )V3T6 T7<latexit sha1_base64="Pja/6IRGgik86FHPrvK0hjPeN9w=">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</latexit><latexit sha1_base64="DXfzMfQeC50gpronrLctQJiz5vw=">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</latexit>

T5 = T3X

T5 = T3Xc

T5 = T3X

41/66

c“Converse” result (II)
1
1
1

+ (1

+2(1
T3 )V
T3 )V2 1

T3 + (11

T3 )V2+2 (1TT33)V
T3 )V2
T5 + (1T5 )V31
T3 + (1T3 )V2
T1
+
(1
)V
+3 (1
T5
T5 +3(1T5 + (1TT55)V
T5 )V3T3 + (1T3 )V2
+
(1
+
(1
)V
T3
2 1
T3
(1
3
T5 + T
T5 )V3
T5 + (1T5 )V3(1(1 T3T)V+)V
U 1 T1U1 T1 T3T+
1
1
3
32 2
T5 + (1T5
T3 + (1
T31)V2
+
(1
)V
+
(1
)V
2
2
T
T
T
T
3
3
3
3
1
U 1 T1
T5 + (1T5 )V3
T5 + (1T5 )V3+ (1T3 + (1T3 )V2T)V
T3 )V2(1(1 T5T)V
+
)V
T5T+
1
3 1
UT
+
(1
12 T
T
5
53 3
+
(1
)V
1 (1
2
T
T2
+
)V
)V
3
3
3
T
+
(1
)V
U 1 T1
2
T3 + (1 T
T
3
T
T
5
5
3
5
U52 T4
U1U1T1T1T + (1T3 )V2
3
T51+ (1T5 )V3
T5 + (1T5 )V3)V+1 (1+11(1TT2 )V
T2 + (1TT22
T2 )VT2 )V1UU
1 T1
T + (1 3
T5 )V31
V1 1
U2 T4
UT25 T+4 (1 5 UT25 )V
U
T4
2 +
T4(1
)V
1
T
T
2
U 1 T1 U
U1 T12
T2 2
+
(1
)V
3
T
T
2 T4
ϕ
5
5
U 1 T1U2 TT
t+ (1T2 )V1T3 + (1T3 )V2
1 T
24
TU
1 11 1T2 + (1T )V
T2 + (1T2 )V1U2 T4
U 1 2 T1
(1(1 T2T)V
U2 T4
+
)V U
T2T+
2
2 1 1 2 T4+ (1T2 )V1
+
(1
)V
(1
)V
1
T2 + (1T2 )V1
T
T
3
T
T
U52U+
2
5
2T 2T
T6 64 4UT2 T43
U2 T4 + (1
U2 T4
(1
3
T2 +
T2 )V1
T2
Tc2 )V1cc
X (1U 1 T1
12 = TX
T2T+
T2 )VT
T4 U=2 TT2X
T3 = T2X
T5 = T3X
1 =T2 +T(1T2 )V1U2 T41
1
4U
2 T4
T4 4 U2c T
+
(1
)V
1
T
T
4
c
c
c
c
c
c
c
c
c
c
2
2
X
XT X
5 T T=
5
T4T=
T1 = TT1X=2XT2X=1X
T X T3 =
TU=
TT1X
T2X
T5 = T3X
T31X = T15X
T4 = T2X
T4 = T2X
T1X
T3 T=4 T
T5 = T3X
T2 T=3 T
TT3X
5 =cT
22 =
2 3 =
2
2
c 2
4
X
c
c
c
T4T= =
T2X
T3 = T2X
3 TX
T1X
T4 = T2X
T2 = T1X T5 =
T3T=
T5 = T3X
1
2
c
X
X
Xc
Xc
+
(1
)V
1
T
T
T1 = T1X c
T4 = T2c
2 X Xc2
T2 X=cT1
T3 = T2
T5 = T3Xcc
T4 T=2 T
T1T=
T1T2
T2 = T1X=2XT1X
T3 =UT22 T4c c TT45==TT2X
T5 = T3X
3 =3 c c
X cX c
X
X
X
X
T1T=
T1X
T4T=
Tc2X
TX
T1X
T2X
T
=
T
=
T
=
T
=
T
=
T
T
=
T
c T3T=
2T=
5
1
4
2
3
5
c
c
c
c
3
1
2
X
1
2
3
Xc X
Xc
X
T14X= T2 T3 = T2X
T2 =T3 = T2 T2 = T
T1T=
T4 = T2 T5 = T3
T1 = T1X
T4 = T2X
T5 = T3X
T2 = T1X
T3 = T2X
1 T1
c
c
c
c
c
X Xc
T1 = T1X
=
T2X
T4T=
T2X
T2 = XT1X T1 =T3T=
T1XX cT3 = T2X X
T3X
T5 = T3X
5 =1c T2
X c T4T2
Xc
c=c
X
T1 = T1XT1 = T1
2 4 = T2 T4 = T2
T2 = T1XT2 = TT13 = T2XT3 = TT
T5 = T3XT5 = T3
X
X
Xc
Xc
Xc
T1 = T1
T4 = T2
T2 = T1
T3 = T2
T5 = T3

3

T3 + (1T3 + (1T3 )V2

1 1
T3 )V2

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U

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U

U<latexit sha1_base64="LsxKoZ3ytmV5ml1dk6yxYiMjACY=">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</latexit><latexit sha1_base64="qr2hWjwEUOI4JkONUXVF86o4RJM=">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</latexit>

T1 T2 T3
<latexit sha1_base64="8TBF12fPSOyMG2XmMan/CjOaXxg=">AAACx3icjVHLSsNAFD2Nr1pfVZdugkVwVRIRdFl0o7sKfUEtJUmnbWiaCZNJsRQX/oBb/TPxD/QvvDNOQS2iE5KcOfeeM3Pv9ZMoTKXjvOaspeWV1bX8emFjc2t7p7i710h5JgJWD3jERcv3UhaFMavLUEaslQjmjf2INf3RpYo3J0ykIY9rcpqwztgbxGE/DDypqFr3pNAtlpyyo5e9CFwDSjCryosvuEUPHAEyjMEQQxKO4CGlpw0XDhLiOpgRJwiFOs5wjwJpM8pilOERO6LvgHZtw8a0V56pVgd0SkSvIKWNI9JwyhOE1Wm2jmfaWbG/ec+0p7rblP6+8RoTKzEk9i/dPPO/OlWLRB/nuoaQako0o6oLjEumu6Jubn+pSpJDQpzCPYoLwoFWzvtsa02qa1e99XT8TWcqVu0Dk5vhXd2SBuz+HOciaJyUXafs3pyWKhdm1Hkc4BDHNM8zVHCFKurkPcQjnvBsXVvcmlh3n6lWzmj28W1ZDx/4/ZAS</latexit>

T4<latexit sha1_base64="rbdbXszceVlCarz7ZPU3J4CJ62o=">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</latexit><latexit sha1_base64="yi5c0rp6wL2fELOTzUi9HZWJMmM=">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</latexit><latexit sha1_base64="zeO+XWYiVUNpTYsT6d9Biy4r/xY=">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</latexit>

T1 = T1X

T2 = T1Xc

T3 = T2Xc

T4 = T2X

T5<latexit sha1_base64="yI/kTzsucDEwDJpFXLuQhMB4Swo=">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</latexit>

T6 T7<latexit sha1_base64="Pja/6IRGgik86FHPrvK0hjPeN9w=">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</latexit><latexit sha1_base64="DXfzMfQeC50gpronrLctQJiz5vw=">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</latexit>

T5 = T3X

c

41/66

Sketch of the proof

1

Let Qpds, dθq “ÿżδTn psqδpλTn Un q pθq `ně1R` ˆR`

1tθěλs u Q̂pds, dθq.

żtż
λ2 Show that Nt “
03R`

1tθďλs u Qpds, dθq.

Show that Q is a space time Poisson measure of mean measureds b dθ.

42/66

Outline

1 Some generalities on point processes
2 Pathwise representation of point processes in random environment
3 Strong comparison of point processes
Point processes with ordered intensities
General case
Converse result
4 Birth-Death-Swap process in random environment

43/66

A standard nonexplosion criterion
Existence of non-exploding solution to:
żtż
Ntα “
0

R`

1tθďαps,rN α ss ´ qu Qpds, dθq,

dNtα “ Qpdt, s0, αpt, rN α st ´ qsq?

A standard assumption§ Sublinear growth:αpω, s, rnsq ď c ` bnpsq
ş§ Example: Linear Hawkes process (αps, rnsq “ b ` 0s hps ´ r qdnpsq)
with bounded h.
§ Under this assumption:żtżtżt
ErNtα s “ Er αps, rN α ss ´ qdss ď Er pc`bNsα´ qdss “ ct` bErNsα´ sds.
0Gronwall lemma: ErNtα s ď ce bt .

0

0

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Strong comparison of point processes§ Issue: domination by deterministic function, linear assumption canbe relaxed (Markov case).
§ Existence, uniqueness and nonexplosion by strong domination.
§ Comparison of counting processes with ordered intensity processes:
Preston (1975), Bhaskaran (1986), Rolski and Szekli (1991), Bezborodov(2015)).

§ Pathwise representation: well-adapted to study this problem,
simplifies proofs significantly, “strong” pathwise comparison.

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Outline
1 Some generalities on point processes
2 Pathwise representation of point processes in random environment
Process with bounded intensity
General caseA first “converse result”
3 Strong comparison of point processes
Point processes with ordered intensities
General case
Converse result
4 Birth-Death-Swap process in random environment

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Example NHPs with exponential intensities
λ2 ptq “ e ´0.2t ď λ1 ptq “ e ´0.1t .

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Example NHPs with exponential intensities
λ2 ptq “ e ´0.2t ď λ1 ptq “ e ´0.1t .

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Example NHPs with exponential intensities
λ2 ptq “ e ´0.2t ď λ1 ptq “ e ´0.1t .

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Example NHPs with exponential intensities
λ2 ptq “ e ´0.2t ď λ1 ptq “ e ´0.1t .

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Example NHPs with exponential intensities
λ2 ptq “ e ´0.2t ď λ1 ptq “ e ´0.1t .

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Counting processes with ordered intensity
Let N 1 and N 2 two counting processes with ordered pGt q-intensities:
λ2t ď λ1tt ě 0,and defined by:
żtż
Nti “
0

R`

1tuďλis u Qpds, dθq,

§ tu ď λ2s u Ă tu ď λ1s u so that Nt2 ď Nt1 ,i “ 1, 2.
@t ě 0.

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Counting processes with ordered intensity
Let N 1 and N 2 two counting processes with ordered pGt q-intensities:
λ2t ď λ1tt ě 0,and defined by:
żtż
Nti “
0

R`

1tuďλis u Qpds, dθq,

§ tu ď λ2s u Ă tu ď λ1s u so that Nt2 ď Nt1 ,i “ 1, 2.
@t ě 0.

Stronger result§ Let Q 1 pds, dθq “ 1tuďλ1 u Qpds, dθq “ÿδTn1 psqδθn pθq,

s

ně1

żtż

żtż
Nt1 “

Q 1 pds, dθq,
0

Nt2 “

ř

R`

Nt2 “

R`

0

ně1 1tTn1 ďtu 1tθn ďλ2Tn u All jumps of N

21tuďλ2s u Q 1 pds, dθq.

are jumps of N 1
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Strong domination of space-time point
processes§ A space-time point process N 2 is strongly dominated by
space-time point process N 1 , N 2 ă N 1 if
N 1 ´ N 2 is a space-time point process.
§ Example of counting processes:
N 2 ă N 1 ô all jumps of N 2 are jumps of N 1 .
§ Property If two counting processes N 1 and N 2 have orderedintensities and are generated with the same Poisson measure, thenN 2 ă N 1.
Difficulty:§ When λt “ αpω, t, rNst ´ q: natural order of random intensity
functionals does not necessary imply an order on intensities
processes.
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Outline
1 Some generalities on point processes
2 Pathwise representation of point processes in random environment
Process with bounded intensity
General caseA first “converse result”
3 Strong comparison of point processes
Point processes with ordered intensities
General case
Converse result
4 Birth-Death-Swap process in random environment

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Strong order of intensity functionals§ Let α, β : Ω ˆ R` ˆ A Ñ R` two pGt q-predictable functionals.
α is strongly majorized by β, α ďs β if@t ě 0, sup αpt, rmsq ď βpt, rnsq a.s.

(6)rmsărnsş§ Example αpt, rnsq “ λt ` 0t h1 pt ´ sqdnpsq andşt
αpt, rnsq “ λt ` 0 h2 pt ´ sqdnpsq with 0 ď h1 ď h2 .

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Construction by strong domination (1D case)

Proposition (El Karoui, K.)
Assume that there exists a unique well-defined solution N β of:dNtβ “ Qpdt, s0, βpt, rN β st´ qsq.
(i) Then, for all α ďs β, there exists a unique solution of the SDE`
˘
dNtα “ Q dt, s0, αpt, rN α st´ qs .

(7)(ii) Furthermore, N α is strongly dominated by N β (N α ă N β ), i.e.
N β ´ N α is a counting process (jump times of N α “ jump times of N β ).

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Application to nonexplosion criteria (I)α ďs β ñ
żtż
Ntα “
0

R`

1θďαpt,rN α st´ q Qpds, dθq ă Ntβ “

żtż
0

R`

1θďβpt,rN β st´ q Qpds, dθq§ Pathwise comparison result: Ntα ď Ntβ , for f nonnegative functionżtżtαf psqdNs ďf psqdNsβ ,...
0

0

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Application to nonexplosion criteria (I)α ďs β ñ
żtż
Ntα “
0

R`

1θďαpt,rN α st´ q Qpds, dθq ă Ntβ “

żtż
0

R`

1θďβpt,rN β st´ q Qpds, dθq§ Pathwise comparison result: Ntα ď Ntβ , for f nonnegative functionżtżtαf psqdNs ďf psqdNsβ ,...
00

§ A first application: case βpt, rnsq “ c ` bnptq
N β is a Markov birth process of pGt q-intensity pc ` bNtβ q. Its
distribution does not depends on Q.
α ďs β corresponds to Sublinear growth condition.
ñ Ntα ď Ntβ ă 8, for all t ě 0.

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Application to nonexplosion criteria (II)Goal Define a class of dominating intensity functionals β associated withnonexploding processes.
§ Preceding proof does not rely on the linearity ofβpt, rnsq “ c ` bnptq.
§ Extension to nonlinear dominating intensities
Markov birth processes (βpt, rnsq “ g pnptqqq): n.s.c fornonexplosion

ÿ 1
“ 8.
g pjq
j“0

(8)

If α ďs g with g verifying (8), unique nonexploding solution
N α of (7).
Sometimes known as the Jacobsen condition (proof without
pathwise representation much harder!).
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Application to nonexplosion criteria (III)

Goal Define a class of dominating intensity functionals β associated withnonexploding processes.
§ A second application: if α ďs pλt q withşt0

λs ds ă 8, thenαunique nonexploding solution N of (7).

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Application to nonexplosion criteria (III)

Goal Define a class of dominating intensity functionals β associated withnonexploding processes.
§ A second application: if α ďs pλt q withşt0

λs ds ă 8, thenαunique nonexploding solution N of (7).
§ Corollary of the proposition
If α ďs β with βpt, rnsq “ kt g pnptqq, pkt q predictable locallyř 1bounded process and g verifying
g pjq , then (7) admits a uniquewell-defined solution N α .

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Other following results§ Corollary 2: @ α ďs β, the sequence of jump times of N β is aştlocalizing sequence of the local martingales Ntα ´ 0 αps, Nsα qds.
§ Corollary 3: Let pαi qiPI be a family of intensity functionals withαi ďs β and:dNti “ Qpdt, s0, αi pt, rNst ´ qsq
Then pN i qiPI is tight.
§ Straigthforward extension to strong comparison for space-time point
processes.

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Sketch of the proofα ďs β = N β ,dNtα “ Qpdt, s0, αpω, t, Ntα´ qsq ?

(9)§ Step 1: replace Q by
Q β pdt, dθq “ 1tθďβpt,rN β st ´ qu Qpdt, dθq “ř

ně1 δTnβ pdtqδθn pdsq

d Ñtα “ Q β pdt, s0, αpω, t, Ñtα´ qsq.

(10)

Jumps times of Q g can be enumerated increasingly ñ (10).
§ Step 2: show equality between (9) and (10).
Remark The proof also gives a simulation algorithm.

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Outline
1 Some generalities on point processes
2 Pathwise representation of point processes in random environment
Process with bounded intensity
General caseA first “converse result”
3 Strong comparison of point processes
Point processes with ordered intensities
General case
Converse result
4 Birth-Death-Swap process in random environment

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Converse result (I)Let N λ withpGt q- intensity pλt q and such that N α ă N β .
Is there a representation of N λ in terms of stochastic integral withrespect to a marked process with same jumps than N β ?
§ YES§ Ingredients:
A sequence of i.i.d variables pUn qně0 uniform on r0, 1s andindependent of G8 .
A sequence of i.i.d variables pVn qně0 uniform on r0, 1s andindependent of G8 .

§ See also Rolski and Szekli (1991) (distributional viewpoint)

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Converse result (II)Notations:
X “ N α,X c “ N β ´ N α,

φt “

λt
λβt

ď 1.

1
T3 + (1T3 )V2
T5 + (1U1

T5 )V3T1

T2 + (1T1 = T1X

ϕt

T2 )V1T2 = T1Xc

T3 = T2X

U2

c

T4

T4 = T2X

T5 = T3X

c

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Converse result (II)Notations:
X “ N α,X c “ N β ´ N α,

φt “

λt
λβt

ď 1.

1
T3 + (1T3 )V2
T5 + (1U1

T5 )V3T1

T2 + (1T1 = T1X

ϕt

T2 )V1T2 = T1Xc

T3 = T2X

U2

c

T4

T4 = T2X

T5 = T3X

c

Corollary Two counting processes with the same intensity functional andstrongly dominated by the same process have the same distribution.

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Outline

1 Some generalities on point processes
2 Pathwise representation of point processes in random environment
3 Strong comparison of point processes
4 Birth-Death-Swap process in random environment

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Birth Death Swap (BDS) systems
An example of BDS system
Environnement € ,MortNaissance/Arrivée

Sous groupe 1

Swap (3, 2)

Sous groupe 2

Swap (2, 3)

Sous groupe 3§ The variability of the environment is taken into account ñ
stochastic intensities:
Pp ev of type γ Pst, t ` dts|Gt q » µγ pω, t, Zt qdt.

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References

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Some additional references (I)[1] Brémaud, Pierre. Point processes and queues: martingale
dynamics. Vol. 50. New York: Springer-Verlag, 1981.
[2] Cinlar, Erhan. Probability and stochastics. Springer Science &Business Media, 2011.
[3] Daley, Daryl J., and David Vere-Jones. An introduction to thetheory of point processes. Springer Science & Business Media, 2007.
[4] Garcia, Nancy L., and Kurtz, Thomas G. Spatial point processes
and the projection method, In and Out of Equilibrium 2. Birkhäuser
Basel, 2008. 271-298.

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Some additional references (II)[5] Jacod, Jean. Multivariate point processes: predictable projection,
Radon-Nikodym derivatives, representation of martingales,
Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete
31.3 (1975): 235-253.
[6] Kaakaı̈, S. and El Karoui, Nicole. Birth Death Swap population in
random environment and aggregation with two timescales,
arXiv:1803.00790, 2020.
[7] Kallenberg, Olav. Random measures, theory and applications.
Springer International Publishing, 2017.
[8] Massoulié, Laurent. Stability results for a general class ofinteracting point processes dynamics, and applications, Stochastic
processes and their applications 75.1 (1998): 1-30.
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